The postulates and the relativity of simultaneity

Michelson's interferometer splits a beam into two perpendicular arms of length L and recombines them. If light rode an ether through which Earth moved at v ≈ 30 km s⁻¹, the along-wind arm's round trip would take L/(c − v) + L/(c + v) = (2L/c)γ² and the across-wind arm's 2L/√(c² − v²) = (2L/c)γ, with γ = 1/√(1 − v²/c²). Expanding both, the arms differ by Δt ≈ Lv²/c³. Rotating the apparatus by 90° swaps the roles, so the optical path difference swings by 2cΔt and the pattern walks by N = 2Lv²/(λc²) fringes. For the 1887 instrument, L = 11 m (folded by mirrors), λ = 589 nm and (v/c)² = 1.0 × 10⁻⁸: N = 2 × 11 × 10⁻⁸ ÷ 5.89 × 10⁻⁷ = 0.37 fringe, against a resolution near 0.01. They saw under 0.02, and every more sensitive repetition since has seen nothing. Note the scale of the achievement: the interferometer turns a time difference of 3.7 × 10⁻¹⁶ s into a path difference of 110 nm and compares it with a wavelength, which is how a 10⁻⁸ effect became measurable at all.

Einstein kept the null result and dropped the ether. Postulate 1: the laws of physics take the same form in every inertial frame. Galileo had claimed this for mechanics; extending it to electromagnetism means no experiment, optical or electrical, can detect a laboratory's uniform motion, so there is no ether wind to find. Postulate 2: light in vacuum travels at c in every inertial frame, whatever the motion of its source. Maxwell's equations already contain c = 1/√(ε₀μ₀); the postulate says that number is not tied to a medium but is the same for every inertial observer. Each postulate closes a door. The first removes any preferred frame. The second kills Galilean velocity addition for light: a lamp on a train moving at v does not send light forward at c + v as the platform measures it. Since 1983 the SI has built the second postulate in, defining the metre by fixing c = 299 792 458 m s⁻¹ exactly. Both postulates speak of inertial frames only: they constrain observers in uniform relative motion and say nothing about accelerated frames or gravitation.

'Simultaneous' is not given; it has to be constructed. Two identical clocks at rest in a frame, A at x = 0 and B a distance L away, are synchronised by Einstein's recipe. A sends a flash at its reading t₁; the flash reflects at B and returns to A at t₂. A alone can compute L = c(t₂ − t₁)/2 and the reflection time (t₁ + t₂)/2, and B is set to read that value at the moment the flash arrived. With t₁ = 0 and t₂ = 4.00 μs, B sits 600 m away and is set to 2.00 μs. The recipe uses one clock plus the convention that the outward and return legs take equal times. Repeated for every clock in the frame, it defines the frame's time coordinate, and two events are simultaneous in that frame when the clocks at their locations agree. Postulate 2 makes the recipe consistent: synchronising B from A and then C from B agrees with synchronising C from A directly. Postulate 1 grants every inertial frame the right to run the same recipe with its own clocks. Disagreement is now unavoidable, because each frame trusts light to move at c relative to itself.

Platform frame S; a train of S-length L = 300 m moves at v = 0.60c, so γ = 1.25. Lightning strikes both ends at t = 0 in S, meaning S's clocks at x = 0 and x = 300 m both read zero. A passenger sits at the train's midpoint. In S the front flash travels backward while she moves forward, so they close at c + v and meet after L/(2(c + v)) = 300 m ÷ (2 × 1.6c) = 312.5 ns; the rear flash chases her at closing speed c − v and arrives after L/(2(c − v)) = 1250 ns. Closing speeds are legal in S: the light still moves at c, and the passenger moves too. She receives the front flash 937.5 ns of S-time before the rear one. Now switch frames. In S′ she is at rest at the midpoint of her own train, both flashes travelled equal distances at c by postulate 2, and they arrived at different times. The only conclusion open to her is that the front strike happened first. Her clocks at the two ends of the train, synchronised by the light recipe in S′, record exactly that. Neither frame is wrong; each has applied the same definition of simultaneity to its own clocks.

The train gives the rule. For two events simultaneous in S and separated by Δx along the direction of S′'s motion, Δt′ = t′₂ − t′₁ = −γvΔx/c², where event 2 lies ahead in the direction S′ travels. The thought experiment supplies the leading-order form −vΔx/c² directly; the exact factor γ enters because the moving frame's rods and clocks are themselves rescaled, which the next two topics derive, and the Lorentz transformation confirms the full formula. The numbers agree: the 937.5 ns arrival gap divided by γ is 750 ns, and γβL/c = 1.25 × 0.60 × 300 m ÷ (3.00 × 10⁸ m s⁻¹) = 750 ns. Check dimensions: (m s⁻¹)(m)/(m² s⁻²) = s. Check the limit: as v/c → 0, γ → 1 and Δt′ → −vΔx/c² → 0, Galileo's shared time. Check the size: a car at 30 m s⁻¹ passing two flashes 100 m apart finds them 3.3 × 10⁻¹⁴ s apart, below any clock. Two rules follow. Events at one place (Δx = 0) that are simultaneous in one frame are simultaneous in all. Separation across the motion never enters; only the component along v matters.

The postulates constrain inertial frames in flat spacetime, and only those. They do not say accelerated frames are equivalent to inertial ones; an accelerating body is described from an inertial frame, and the passenger who feels the train brake knows she is not inertial, whatever the postulates say about uniform motion. They also say nothing about gravitation. A clock higher in a gravitational potential runs faster than one below, an effect outside special relativity that general relativity supplies; GPS satellite clocks gain about 45 μs per day from altitude and lose about 7 μs per day from speed, and both corrections are applied. Finally, the postulates make c a property of spacetime rather than of light. It is the one speed every inertial observer agrees on, and light travels at it because the photon is massless; had the photon a small mass, the invariant speed would still exist and light would fall short of it. Everything that follows in this section, dilation, contraction, the Lorentz transformation and E = γmc², is derived from these two sentences and inherits their domain.